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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

In an arms race between two superpowers, each nation takes stock of its own and its enemy's nuclear arsenal each year. Each nation has the policy of dismantling a certain percentage of its stockpile each year and adding that same percentage of its competitor's stockpile. Nation A dismantles 20%,
Joe has $3.30 in his pocket, made up of nickels, dimes, and quarters. There are 30 coins, and there are five times as many dimes as quarters. How many quarters does Joe have?
Identify each statement as true or false. (a) If a system of linear equations has two different solutions, it must have infinitely many solutions. (b) If a system of linear equations has more equations than variables, it cannot have a unique solution. (c) If a system of linear equations has more
Identify each statement as true or false. (a) Every matrix can be added to itself. (b) Every matrix can be multiplied by itself.
1. Make up a system of two linear equations, with two variables, that has infinitely many solutions. 2. Make up a system of two linear equations, with two variables, that has no solution. 3. If the product of two numbers is zero, then one of the numbers must be zero. Make up two 2 × 2 matrices A
Use the Gauss-Jordan elimination method to find all solutions of the system of linear equations.1.2. 3.
Why should the numbers in a single column of an input-output matrix have a sum that is less than 1?
Perform the indicated matrix operation.1.2. 3. 4.
State whether the inequality is true or false. 1. 2 ≤ - 3 2. -2 ≤ 0 3. 7 ≤ 7 4. 0 ≥ 1/2
Determine whether or not the given point satisfies the given inequality. 1. 3x + 5y ≤ 12, (2, 1) 2. -2x + y ≥ 9, (3, 15) 3. y ≥ - 2x + 7, (3, 0)
Graph the given inequality by crossing out (i.e., discarding) the points not satisfying the inequality.1. y ‰¤ 1/3x + 12. y ‰¥ - x + 1 3. x ‰¥ 4 4. y ‰¤ 2
Give the linear inequality corresponding to the graph.1.2.
Graph the given inequality. 1. y ≤ 2x + 1 2. y ≥ - 3x + 6 3. x ≥ 2 4. x ≥ 0
Graph the feasible set for the system of inequalities.1.2.
Determine whether the given point is in the feasible set of this system of inequalities:1. (8, 7) 2. (14, 3) 3. (9, 10) 4. (16, 0)
Solve for x. 1. 2x - 5 ≥ 3 2. 3x - 7 ≤ 2 3. - 5x + 13 ≤ - 2
Determine whether the given point is above or below the given line. 1. y = 2x + 5, (3, 9) 2. 3x - y = 4, (2, 3) 3. 7 - 4x + 5y = 0, (0, 0) 4. x = 2y + 5, (6, 1)
1. Give a system of inequalities for which the graph is the region between the pair of lines 8x - 4y - 4 = 0 and 8x - 4y = 0.2. The shaded region in Fig. 9 is bounded by four straight lines. Which of the following is not an equation of one of the boundary lines?(a) y = 0(b) y = 2(c) x = 0(d) 2x +
Graph the line 4x - 2y = 7. (a) Locate the point on the line with x-coordinate 3.6. (b) Does the point (3.6, 3.5) lie above or below the line? Explain.
Graph the line x + 2y = 11. (a) Locate the point on the line with x-coordinate 6. (b) Does the point (6, 2.6) lie above or below the line? Explain.
Display the feasible set in Exercise 47.In exerciseGraph the feasible set for the system of inequalities.
Display the feasible set in Exercise 48.In exerciseGraph the feasible set for the system of inequalities.
Write the linear inequality in slope-intercept or vertical form. 1. 2x + y ≤ 5 2. - 3x + y ≥ 1 3. 5x - 1/3y ≤ 6
Determine whether the given point is in the feasible set of the furniture manufacturing problem. The inequalities are as follows.1. (8, 7) 2. (14, 3) 3. (9, 10) 4. (16, 0)
A coal company owns mines in two different locations. Each day, mine 1 produces 4 tons of anthracite (hard) coal, 4 tons of ordinary coal, and 7 tons of bituminous (soft) coal. Each day, mine 2 produces 10 tons of anthracite, 5 tons of ordinary coal, and 5 tons of bituminous coal. It costs the
A student is taking an exam consisting of 10 essay questions and 50 short-answer questions. They have 90 minutes to take the exam and know they cannot possibly answer every question. The essay questions are worth 20 points each, and the short-answer questions are worth 5 points each. An essay
A local politician has budgeted at most $80,000 for her media campaign. She plans to distribute these funds between TV ads and radio ads. Each one-minute TV ad is expected to be seen by 20,000 viewers, and each one-minute radio ad is expected to be heard by 4000 listeners. Each minute of TV time
A dairy farmer concludes that his small herd of cows will need at least 4550 pounds of protein in their winter feed, at least 26,880 pounds of total digestible nutrients (TDN), and at least 43,200 international units (IUs) of vitamin A. Each pound of alfalfa hay provides .13 pound of protein, .48
A clothing manufacturer makes denim and hooded fleece jackets. Each denim jacket requires 2 labor-hours for cutting the pieces, 2 labor-hours for sewing, and 1 labor-hour for finishing. Each hooded fleece jacket requires 1 labor-hour for cutting, 4 labor-hours for sewing, and 1 labor-hour for
Consider the furniture manufacturing problem discussed in this section. Suppose that the company manufactures only chairs. What is the maximum number of chairs that could be manufactured?
Consider the furniture manufacturing problem discussed in this section. Suppose that the company manufactures only sofas. What is the maximum number of sofas that could be manufactured?
Joe's Confectionary puts together two prepackaged assortments to be given to trick-or-treaters on Halloween. Assortment A contains 2 candy bars and 2 suckers and yields a profit of 40 cents. Assortment B contains 1 candy bar and 2 suckers and yields a profit of 30 cents. The store has available 500
Mr. Holloway decides to feed his pet Siberian husky two dog foods combined to create a nutritious low-sodium diet. Each can of brand A contains 3 units of protein, 1 unit of calories, and 5 units of sodium. Each can of brand B contains 1 unit of protein, 1 unit of calories, and 4 units of sodium.
A truck traveling from New York to Baltimore is to be loaded with two types of cargo. Each crate of cargo A is 4 cubic feet in volume, weighs 100 pounds, and earns $13 for the driver. Each crate of cargo B is 3 cubic feet in volume, weighs 200 pounds, and earns $9 for the driver. The truck can
Determine x and y so that the objective function 4x + 3y is maximized.1.2.
Consider the nutrition problem of Example 1. Suppose that the only food available was rice. How many cups of rice would be required to meet the nutritional requirements? Suppose that a person decides to make rice and soybeans part of their staple diet. The object is to design a lowest-cost diet
Consider the nutrition problem of Example 1. Suppose that the only food available was soybeans. How many cups of soybeans would be required to meet the nutritional requirements? Suppose that a person decides to make rice and soybeans part of their staple diet. The object is to design a lowest-cost
Refer to Exercises 3.2, Problem 7. How many of each assortment should be prepared in order to maximize profits? What is the maximum profit? (See the graph of the feasible set in Fig. 15.)In problem 2x + y
Refer to Exercises 3.2, Problem 8. How many cans of each dog food should he give to his dog each day to provide the minimum requirements with the least amount of sodium? What is the least amount of sodium? (See the graph of the feasible set in Fig. 16.)In problem 3 - x - y
Refer to Exercises 3.2, Problem 9. How many crates of each cargo should be shipped in order to satisfy the shipping requirements and yield the greatest earnings? (See the graph of the feasible set in Fig. 17.)In problem 8x + y
Refer to Exercises 3.2, Problem 10. Find the number of days that each mine should be operated in order to fill the order at the least cost. (See the graph of the feasible set in Fig. 18.)In problem 3x + 2y
Refer to Exercises 3.2, Problem 11. How many of each type of question should the student do to maximize the total score? (See the graph of the feasible set in Fig. 19.)In problem 2x + 3y
Refer to Exercises 3.2, Problem 12. How should the media funds be allocated so as to maximize the total audience? (See the graph of the feasible set in Fig. 20.)In problem x + 8y
Refer to Exercises 3.2, Problem 13. How many pounds of each food should be purchased in order to meet the nutritional requirements at the least cost? (See the graph of the feasible set in Fig. 21.)
Refer to Exercises 3.2, Problem 14. How many of each type of jacket should be made to maximize the profit? (See the graph of the feasible set in Fig. 22.)
Rework the furniture manufacturing problem, where everything is the same except that the profit per chair is changed to the given value. (See Table 1 for vertices.) 1. $150 2. $60
Minimize the objective function 3x + 4y subject to the constraints
Maximize the objective function 7x + 4y subject to the constraints
Maximize the objective function 2x + 5y subject to the constraints
Minimize the objective function 2x + 3y subject to the constraints
Maximize the objective function 100x + 150y subject to the constraints
Minimize the objective function 1/2x + 3/4y subject to the constraints
Minimize the objective function 7x + 4y subject to the constraints
Maximize the objective function x + 2y subject to the constraints
Infotron, Inc., makes electronic hockey and soccer games. Each hockey game requires 2 labor-hours of assembly and 2 labor-hours of testing. Each soccer game requires 3 labor-hours of assembly and 1 labor-hour of testing. Each day, there are 42 labor-hours available for assembly and 26 labor-hours
An electronics company has factories in Cleveland and Toledo that manufacture Bluray and DVD players. Each day, the Cleveland factory produces 500 Blu-ray and 300 DVD players at a cost of $18,000. Each day, the Toledo factory produces 300 of each type of player at a cost of $15,000. An order is
A farmer has 100 acres on which to plant oats or corn. Each acre of oats requires $18 capital and 2 hours of labor. Each acre of corn requires $36 capital and 6 hours of labor. Labor costs are $8 per hour. The farmer has $2100 available for capital and $2400 available for labor. If the revenue is
A company makes two items, I1 and I2, from three raw materials, M1, M2, and M3. Item I1 uses 3 ounces of M1, 2 ounces of M2, and 2 ounces of M3. Item I2 uses 4 ounces of M1, 1 ounce of M2, and 3 ounces of M3. The profit on item I1 is $8 and on item I2 is $6. The company has a daily supply of 40
The E-JEM Company produces two types of laptop computer bags. The regular version requires $32 in capital and 4 hours of labor and sells for $46. The deluxe version requires $38 in capital and 6 hours of labor and sells for $55. How many of each type of bag should the company produce in order to
A refinery has two smelters that extract metallic iron from iron ore. Smelter A processes 1000 tons of iron ore per hour and uses 7 megawatts of energy per hour. Smelter B processes 2000 tons of iron ore per hour and uses 13 megawatts of energy per hour. Each refinery must be operated at least 8
A nutritionist, working for NASA, must meet certain nutritional requirements for astronauts and yet keep the weight of the food at a minimum. They are considering a combination of two foods, which are packaged in tubes. Each tube of food A contains 4 units of protein, 2 units of carbohydrates, and
A contractor builds two types of homes. The first type requires one lot, $12,000 capital, and 150 labor-days to build and is sold for a profit of $2400. The second type of home requires one lot, $32,000 capital, and 200 labor-days to build and is sold for a profit of $3400. The contractor owns 150
The Beautiful Day Fruit Juice Company makes two varieties of fruit drink. The given chart shows the composition and profit per can for each variety. Each week, the company has available 9000 ounces of pineapple juice, 2400 ounces of orange juice, and 1400 ounces of apricot juice. How many cans of
The Bluejay Lacrosse Stick Company makes two kinds of lacrosse sticks. The accompanying chart shows the labor requirements and profits for each type of lacrosse stick. Each day, the company has available 120 labor-hours for cutting, 150 labor-hours for stringing, and 140 labor-hours for finishing.
Suppose that the farmer of Exercise 35 can allocate the $4500 available for capital and labor however he wants to. (a) Without solving the linear programming problem, explain why the optimal profit cannot be less than that found in Exercise 35. (b) Find the optimal solution in the new situation.
Pavan wants to add a sliced carrot and green pepper salad to his dinner each day to help him meet some of his nutritional needs. The nutritional information and costs per cup are contained in the given chart. He would like for his salad to contain at least 15 g of fiber, 100 mg of calcium, and 100
A small candy shop makes a special Cupid assortment, with 60 red pieces of candy and 40 white pieces of candy, that makes a profit of $8. A special Patriotic assortment has 30 pieces of red candy, 35 pieces of white candy, and 35 pieces of blue candy and makes a profit of $6. How many of each
A portrait studio specializes in family portraits. They offer a Basic package that costs $25 to produce and an Heirloom package that costs $40 to produce. To have a successful week, the studio must sell at least 50 Basic packages at $30 each and at least 34 Heirloom packages at $75 each, with total
A bath shop sells two different gift baskets. The Pamper Me basket contains 1 bottle of shower gel, 2 bottles of bubble bath, and 2 candles and makes a profit of $15. The Best Friends basket contains 2 bottles of shower gel and 2 bottles of bubble bath and makes a profit of $12. Each week, the shop
A florist offers two types of Thank You bouquets. The Thanks a Bunch bouquet consists of 3 roses, 4 carnations, and 2 stems of baby's breath and sells for $12. The Merci Beaucoup bouquet consists of 6 roses, 3 carnations, and 2 stems of baby's breath and sells for $16. The florist has 48 roses, 34
Consider the following linear programming problem: Minimize M = 10x + 6y subject to the constraintsDetermine a point of the feasible set.
Find the values of x and y that maximize the given objective function for the feasible set in Fig. 13.1. x + 2y 2. x + y
Consider the following linear programming problem: Maximize M = 10x + 6y subject to the constraints(a) Sketch the feasible set. (b) Determine three points in the feasible set, and calculate M at each of them. (c) Show that the objective function attains no maximum value for points in the feasible
1. Use Excel or Wolfram0 Alpha to solve Exercise 25.2. Use Excel or Wolfram0Alpha to solve Exercise 26.
Find the values of x and y that minimize the given objective function for the feasible set in Fig. 14.1. 8x + y 2. 3x + 2y
Figure 10(a) shows the feasible set of the nutrition problem in Example 1 of Section 3.3 and the straight line of all combinations of rice and soybeans for which the cost is 42 cents. (a) The objective function is 21x + 14y. Give the linear equation (in slope-intercept form) of the line of constant
Consider the feasible set in Fig. 13. For what values of k will the objective function x + ky be maximized at the vertex (3, 4)?
Consider the feasible set in Fig. 14. Explain why the objective function ax + by, with a and b positive, must have its maximum value at point E.
Mr. Smith decides to feed his pet Doberman pinscher a combination of two dog foods. Each can of brand A contains 3 units of protein, 1 unit of carbohydrates, and 2 units of fat and costs 80 cents. Each can of brand B contains 1 unit of protein, 1 unit of carbohydrates, and 6 units of fat and costs
An oil company owns two refineries. Refinery I produces each day 100 barrels of high-grade oil, 200 barrels of medium-grade oil, and 300 barrels of low-grade oil and costs $10,000 to operate. Refinery II produces each day 200 barrels of high-grade, 100 barrels of medium-grade, and 200 barrels of
Mr. Jones has $9000 to invest in three types of stocks: low-risk, medium-risk, and high-risk. He invests according to three principles. The amount invested in low-risk stocks will be at most $1000 more than the amount invested in medium-risk stocks. At least $5000 will be invested in low- and
A produce dealer in Florida ships oranges, grapefruits, and avocados to New York by truck. Each truckload consists of 100 crates, of which at least 20 crates must be oranges, at least 10 crates must be grapefruits, at least 30 crates must be avocados, and there must be at least as many crates of
A foreign-car wholesaler with warehouses in New York and Baltimore receives orders from dealers in Philadelphia and Trenton. The dealer in Philadelphia needs 4 cars, and the dealer in Trenton needs 7. The New York warehouse has 6 cars, and the Baltimore warehouse has 8. The cost of shipping cars
Consider the foreign-car wholesaler discussed in Exercise 17. Suppose that the cost of shipping a car from New York to Philadelphia is increased to $110 and all other costs remain the same. How many cars should be shipped from each warehouse to each dealer in order to minimize the cost?
An oil refinery produces gasoline, jet fuel, and diesel fuel. The profits per gallon from the sale of these fuels are $.15, $.12, and $.10, respectively. The refinery has a contract with an airline to deliver a minimum of 20,000 gallons per day of jet fuel and/or gasoline. It has a contract with a
Figure 10(b) shows the feasible set of the transportation problem of Example 2 and the straight line of all combinations of shipments for which the transportation cost is $240.(a) The objective function is [cost] = 375 - 2x - 3y. Give the linear equation (in slope-intercept form) of the line of
Suppose that a price war reduces the profits of gasoline in Exercise 19 to $.05 per gallon and that the profits on jet fuel and diesel fuel are unchanged. How many gallons of each fuel should now be produced to maximize the profit?
A shipping company is buying new trucks. The high-capacity trucks cost $50,000 and hold 320 cases of merchandise. The low-capacity trucks cost $30,000 and hold 200 cases of merchandise. The company has budgeted $1,080,000 for the new trucks and has a maximum of 30 people qualified to drive the
Suppose that the shipping company of Exercise 21 needs to buy enough new trucks to be able to ship 11,200 cases of merchandise. Of course, the company is willing to increase its budget. (a) How many of each type of truck should the company purchase to minimize cost? (b) What if the company hires 23
A major coffee supplier has warehouses in Seattle and San José. The coffee supplier receives orders from coffee retailers in Salt Lake City and Reno. The retailer in Salt Lake City needs 400 pounds of coffee, and the retailer in Reno needs 350 pounds of coffee. The Seattle warehouse has 700
Consider the coffee supplier discussed in Exercise 23. Suppose that the cost of shipping from San José to Reno is increased to $3.00 per pound and all other costs remain the same. How many pounds of coffee should be shipped from each warehouse to each retailer to minimize the cost?
A pet store sells three different starter kits for 10-gallon aquariums. The accompanying chart shows the contents of each kit. The store has 54 filters, 100 pounds of gravel, and 53 packages of fish food available. If the store makes as many of kit I as kits II and III together, how many of each
An automobile manufacturer has assembly plants in Detroit and Cleveland, each of which can assemble cars and trucks. The Detroit plant can assemble at most 800 vehicles in one day at a cost of $1200 per car and $2100 per truck. The Cleveland plant can assemble at most 500 vehicles in one day at a
Refer to Fig. 6. As the lines of constant profit were lowered, the final line had slope - 8/7 and contained the optimal vertex of the feasible set. Figure 15 shows that, as long as the slope of the final line is between -2 and -1, the optimal solution would still be at the same vertex.(a) Suppose
Figure 16 shows the feasible set for the nutrition problem discussed in Example 1 of Section 3.3, along with the purple line of final cost passing through the optimal point. The figure shows that, as long as the slope of the final cost line is between -3 and - 23, the optimal solution will still be
Consider the feasible set in Fig. 17(a). Find an objective function of the form ax + by that has its greatest value at the given point.1. (9, 0) 2. (3, 8)
Consider the feasible set in Fig. 11, where three of the boundary lines are labeled with their slopes. Find the point at which the given objective function has its greatest value.1. 3x + 2y2. 2x + 10y3. 10x + 2y4. 2x + 3y
Consider the feasible set in Fig. 17(b). In Exercises 33-36, find an objective function of the form ax + by that has its least value at the given point.1. (9, 0) 2. (1, 6)
Create a sensitivity report for the nutrition problem of Example 1 of Section 3.3. Use the report to determine the range of optimality for the cost of rice and for the cost of soybeans.
Consider the feasible set in Fig. 12, where three of the boundary lines are labeled with their slopes. In Exercises 7-10, find the point at which the given objective function has its least value.1. 2x + 10y2. 10x + 2y3. 2x + 3y4. 3x + 2y

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